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Equation 22 · Part 5 · The Token Tax of Giving a Model More Tools

Symbol bars

ρ(n,m)  =  1−σ+msˉτ0+nsˉ\rho(n, m) \;=\; 1 - \frac{\sigma + m\bar{s}}{\tau_0 + n\bar{s}}
sˉ\bar{s}

What this part means

bars is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

bars is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

Anthropic’s Tool Search Tool is a direct, documented implementation of that idea. The mechanism is described plainly: “You provide all your tool definitions to the API, but mark tools with deferlr_loading: true to make them discoverable on-demand,” with a lightweight search tool — built-in options include BM25-based and regex-based search — querying an index of names and descriptions, and the full definition of a matching tool injected into context only once the model actually requests it [ 5 ] . The search tool itself is not free, but its overhead is small and fixed rather than proportional to catalogue size: “Only the Tool Search Tool loaded upfront (~500 tokens)” [ 5 ] . Applied to the same…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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